Indeed, the quantification of uncertainty in terms of variance is well adapted when the states are pure. However, this uncertainty includes classical ignorance and a part of quantum nature for mixed states. The mixture of states is responsible for the classical part and the quantum part resulting from the non-commutativity between the studied state and the measured observable. Thus, the variance can have contributions from both quantum and classical sources, i.e., \({\rm Var}\left(\rho \right)\equiv {\rm Var}_{q}\left(\rho \right)+{\rm Var}_{c}\left(\rho \right)\) . There are still several methods to quantify the uncertainty, such as those based on entropy measurements which are widely used as indicators of uncertainty in quantum information theory. This type of measurement is still insufficient and limited because these quantifiers are always influenced by the mixed state of the studied state. In order to determine the quantum part of the variance (REF ), Wigner and Yanase introduced the notion of skew information as [1]}, [2]}
\({\cal I}\left( {\rho ,K} \right): = {{\cal I}_\rho }\left( K \right): = \frac{1}{2}{\rm Tr}\left\lbrace  {{{\left( {i\left[ {\sqrt{\rho },K} \right]} \right)}^2}} \right\rbrace ,\) 
